Inference on the stress strength reliability with exponentiated generalized Marshall Olkin-G distribution

In this paper, an inference on stress-strength reliability model is introduced in case of the exponentiated generalized Marshall Olkin G family of distributions. The maximum likelihood estimator of the stress-strength reliability function is deduced. An asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are derived. A Bayesian inference is introduced for the stress-strength reliability. A simulation is introduced to obtain the maximum likelihood and Bayesian estimates for the stress strength reliability. Real data applications are provided to show the results for the stress-strength model and compare the exponentiated generalized Marshall Olkin-G distribution with other distributions.


Introduction
In life testing, it is very important to study the stress-strength reliability which refers to the quantity R = P (Y < X) and it is assumed that two independent random variables X and Y where X represents the stress of a component and Y represents the strength of the component such that the component fails in case that the stress exceeds the strength.
Al-Mutairi et. al [1] presented an inference on stress-strength reliability from Lindley distribution. Rao et. al [2] introduced an estimation of stress-strength reliability from inverse Rayleigh distribution. Singh et al. [3] presented an estimation of the stress strength reliability subject to the inverted exponential distribution. Al-Mutairi [4] discussed an inference on stress-strength reliability from weighted Lindley distributions. Sharma [5] presented a stressstrength reliability model subject to the inverse Lindley distribution with application to head and neck cancer data.
Mokhlis et al. [6] proposed the stress-strength reliability with general form distributions. Ç etinkaya and Genç [7] introduced the stress-strength reliability estimation under the standard two-sided power distribution. Almarashi [8] introduced an estimation of the stressstrength reliability for Weibull distribution with application. Muhammad et al. [9] proposed an estimation of the reliability of a stress-strength system from Poisson half logistic distribution. Nojosa and Rathie [10] introduced Stress-strength reliability estimation involving generalized gamma and Weibull distributions. Alamri et al. [11] studied the stress-strength reliability where the strength (X) follows Rayleigh-half-normal distribution and stress follows Rayleigh-half-normal distribution, exponential distribution, Rayleigh distribution, and half-normal distribution, respectively. Hassan et al. [12] introduced stress-strength reliability for the generalized inverted exponential distribution using median ranked set sampling. Abu El Azm et al. [13] presented a study for stress-strength reliability subject to exponentiated inverted Weibull distribution with application on breaking of jute fiber and carbon fibers. Jha et al. [14] discussed the multicomponent stress-strength reliability estimation based on unit generalized Rayleigh distribution.
Jha et al. [15] proposed the multicomponent stress-strength reliability estimation based on unit generalized exponential distribution. Maurya et al. [16] introduced a reliability estimation in a multicomponent stress-strength model based on inverse Weibull distribution. Jovanovic et al. [17] proposed an inference on reliability of stress-strength model with Peng-Yan extended Weibull distributions. Sabry et al. [18] presented a Monte Carlo simulation of the stressstrength model and reliability estimation for extension of the exponential distribution. Zarei and Shahrestani [19] proposed the Bayes and empirical Bayes estimator of reliability function in multicomponent stress-strength system based on generalized Rayleigh distribution.
The exponentiated Marshall Olkin family distribution is used in practice since it makes the kurtosis of data more flexible compared to the baseline model and is used to construct heavytailed distributions for modeling real data. Special models can be constructed from the exponentiated Marshall Olkin family distribution with all types of the hazard rate functions. The exponentiated Marshall Olkin family distribution is used to provide consistently better fits than other generated models under the same baseline distribution. The Marshall Olkin family distribution is introduced and discussed in many papers in literature such as [20][21][22][23][24][25][26].
This paper deals with the inference of the stress-strength reliability in case of the stress and strength variables follow the exponentiated generalized Marshall Olkin G family of distributions. The maximum likelihood estimation of the stress-strength reliability is discussed. An asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are discussed. Bayesian inference and the credible interval for the stress-strength reliability function are introduced. A simulation is carried out to obtain the results for the stress strength reliability in case of the exponentiated generalized Marshall Olkin Weibull distribution. Real data applications are introduced to study the goodness of fit of the real datasets.

Exponentiated generalized Marshall Olkin-G distribution
Handique et al. [20] introduced the exponentiated generalized Marshall Olkin G family of distributions (EGMO-G) with cumulative distribution function and probability density function defined as follows where G(x; λ k ) and g(x; λ k ) are the cumulative distribution function and the probability density function of a random variable X, respectively, with parameter vector λ k , k = 1,2,. . .,l.
For example, the cumulative distribution function and the probability density function of a random variable X that follows the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) are given by where a, b, ϑ and γ are the shape parameters and β is the scale parameter. And the cumulative distribution function and the probability density function of a random variable X that follows the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) are given by where a, b, ϑ and γ are the shape parameters and λ is the scale parameter.

Stress-strength reliability
Let X and Y are two independent random variables follow the EGMO-G distribution, then the stress-strength reliability function will be given by Solving this integral, an expression for the stress-strength reliability is obtained as

Maximum likelihood estimation
Assuming that the two independent random samples (X 1 , X 2 ,. . .,X n ) and (Y 1 , Y 2 ,. . .,Y m ) are selected from the EGMO-G distribution with the parameters (b 1 , a, ϑ, λ k ) and (b 2 , a, ϑ, λ k ), respectively. The likelihood function is given as follows The log-likelihood function is obtained as follows The partial derivatives of the log-likelihood function with respect to (b 1 , b 2 , ϑ, a, λ k ) are obtained as follows Equating the partial derivatives to zero and then solving the equations numerically yields the maximum likelihood estimators for the parameters (b 1 , b 2 , ϑ, a, λ k ).
The maximum likelihood estimators for the parameters b 1 and b 2 are obtained aŝ The maximum likelihood estimates of the stress-strength reliability is obtained by substituting in Eq (1).R The partial derivatives with respect to the unknown parameters cannot be solved directly, so a simulation depends on methods like the Newton-Raphson method will be used to obtain the estimates of the unknown parameters and hence calculate the estimate of the stressstrength reliability.

Asymptotic Confidence Interval (A.C.I)
The observed Fisher information matrix of the stress strength reliability parameters is given by The elements of this matrix are obtained by differentiating the Eqs from (3) to (7) with respect to (b 1 , b 2 , ϑ, a, λ k ), respectively, the results are obtained as The asymptotic variances of the parameters (b 1 , b 2 , ϑ, a, λ k ) are given by The asymptotic variance of an estimateR is given by ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi VarðRÞ The asymptotic confidence interval is used for large samples and do not perform well for small samples. So, the bootstrap confidence interval method will be proposed.

Bootstrap Confidence Interval (B.C.I.)
The bootstrap method for constructing confidence intervals is illustrated by the following algorithm.
Step 4: Rearrange the estimates obtained in step (3)

Bayesian estimation
Bayesian estimation method will be discussed into cases. The first case when the parameters are unknown and the second case when the parameters are known.

Case I: When the parameters are unknown
The Bayesian estimator for the stress-strength reliability will be obtained assuming that the parameters b 1 , b 2 , ϑ, a and λ k are independent random variables with prior follow gamma distribution as follows The joint posterior density function of b 1 , b 2 , ϑ, a and λ k given the data (x, y) is given by The marginal posterior distributions of b 1 , b 2 , ϑ, a and λ k can be deduced as It is obvious that seen that posterior samples for b 1 , b 2 , ϑ and a can be generated using gamma distribution. However, λ k cannot be directly simulated from its posterior distribution as it is not in known form and in this case the Metropolis-Hastings algorithm can be applied to simulate random samples from the posterior density of λ k .
The Bayesian estimator of the reliability function R under the squared error loss function using the posterior mean is given by This integral has no analytical solution and the Markov Chain Monte Carlo (MCMC) simulation method can be applied to obtain the Bayesian estimation of the stress-strength reliability. In the method of the Markov chain Monte Carlo, samples are generated from the posterior density function and in turn to compute the Bayes estimates of the reliability function.

MCMC method
Markov chain Monte Carlo (MCMC) method comprises a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis-Hastings algorithm. The steps of applying MCMC method and the Metropolis-Hastings algorithm are presented as follows. Step1: Step 2: Set t =1 Step 3: Generate b ðtÞ 1 from Gamma(n+η 1 , H 1 ) Step 4: Generate b ðtÞ 2 from Gamma(m+η 2 , H 2 ) Step 5: Generate ϑ (t) from Gamma(a(n+m)+η 3 , ξ 3 ) Step 6: Generate a (t) from Gamma(n+m+η 4 ,H 3 ) Step 7: Generate l ðtÞ k from P * 5 ðl k jx; yÞ using MH algorithm as the following ii. Evaluate the acceptance probabilities iii. Generate u k from Uniform(0, 1).
iv. If u k <z k , accept the proposal and set l ðtÞ Step 8: Compute R (t) from Eq (9) Step 9: Set t = t+1 Step 10: Repeat steps from 3 to 9, T times.
Step 11: Stop for sufficiently large value of T, the Bayes estimate of the stress-strength reliability function under the squared error loss will be given aŝ Step 12: To construct the credible interval for R, order R (t) as R (1) <R (2) <� � �<R (T) . Then a 100(1−ε)% credible interval of R becomes ½R B;Tðε=2Þ ;R * B;Tð1À ε=2Þ � Case II: When the parameters are known. In this case the Bayes estimator of the stressstrength reliability under the squared error loss will be given bŷ

PLOS ONE
Inference on the stress strength reliability Let where 0<w 1 <1, w 2 >0. Hence, the Bayes estimator of the stress-strength reliability will be given bŷ The Bayes estimator of the stress strength reliability is deduced aŝ In some situations, it is difficult to find the stress strength reliability from the previous relations and in this case, the MCMC method can be used to find an estimate for the stress strength reliability based on simulated random samples and the steps are given as follows. Step1: Step 2: Set t =1 Step 3: Generate b 1 from Gamma(n+η 1 , H 1 ) Step 4: Generate b 2 from Gamma(m+η 2 , H 2 ) Step 5: Compute R t from Eq (9) Step 6: Set t = t+1 Step 7: Repeat steps from 3 to 6, T times, Step 8: Stop for sufficiently large value of T, the Bayes estimate of the stress-strength reliability function under the squared error loss will be given aŝ

Credible interval
The credible interval for the stress strength reliability can be deduced as follows. From the relations between the gamma distribution and chi-square distribution, it can be shown that The posterior distribution of R can be written as And therefore a 100(1−ε)% credible interval for R will be given by Steps of applying the Monte Carlo simulation is illustrated as follows.
Step 2: Choose the samples sizes (n, m).
Step 3: Generate random values of the random variables X i and Y j at the initial values of the parameters (b 1 , b 2 , ϑ, a, γ, β) by applying the following inversion formula method For EGMO-W distribution the inversion formula will be given by Step 4: Solve the differential Eqs (3)-(5) and using Eqs (8) and (9) to obtain the estimates of the parameters ðb 1 ;b 2 ;Ŵ;â;ĝ;bÞ by using the Newton-Raphson method and the aid of software program R.
Step 5: Obtain the estimate of the stress-strength reliability by substituting in Eq (10).
Step 6: Repeat steps from 3 to 5, 1000 times. In each time of simulation, the same values of the initial parameters and the same samples sizes are considered but the values of generating random samples are varying each time. So, we have 1000 values of estimates of the stressstrength reliabilityR. The mean squared error (MSE) can be obtained from the following relation In Table 1, simulation results are obtained for the stress strength reliability estimates by applying the maximum likelihood method and Bayesian estimation method in case of prior I and prior II when the values of the parameters are given by a = b 1 = b 2 = ϑ = γ = β = 0.5. In Table 2, results for confidence, bootstrap confidence and credible intervals in case of prior I and prior II are deduced when the values of the parameters are given by a = b 1 = b 2 = ϑ = γ = β = 0.5.
In Table 3, simulation results are obtained for the stress strength reliability estimates by applying the maximum likelihood method and Bayesian estimation method in case of prior I and prior II when the values of the parameters are given by a = 0.5, b 1 = 1.5, b 2 = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5. In Table 4, results for confidence, bootstrap confidence and credible intervals in case of prior I and prior II are deduced when the values of the parameters are given by a = 0.5, b 1 = 1.5, b 2 = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5. All results obtained in Tables 1-4 are performed in case of the parameters are unknown (case I) and in case of the parameters are known.
(https://datahub.io/core/ covid-19#resource-covid-19_zip/). It is considered that the stress and strength follow exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) and death rate represents the stress and the recovery rate represents the strength, where Death rate ¼ Deaths Confirmed cases Recovery rate ¼

Recovered cases Confirmed cases
Calculations of recovery and death rates based on the confirmed, recovered cases and deaths of covid 19 data are presented in Table 5. Summary of the measures of the death and recovery rate is illustrated in Table 6.
The goodness of fit for the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) is compared with the generalized Marshall Olkin exponential distribution (GMO-E), the exponentiated generalized exponential distribution (EG-E), the Marshall Olkin exponential distribution (MO-E) and the exponential distribution (E). The results of the goodness of fit for the death and recovery rate of covid 19 data are shown in Table 7.
The best model is chosen as the one having lowest AIC (Akaike Information Criterion). The results obtained in Table 7 indicated that the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) is a better distribution to model the death and recovery rate of covid 19 data than the other distributions. The maximum likelihood estimate of the reliability of the stress-strength model according to the death and recovery rate is obtained as 0.5175 and the 95% asymptotic confidence interval of R is May 2020, the estimated PDF of the death rate, the estimated PDF with histogram of the death rate, the estimated CDF of the death rate and the log likelihood function of the death rate are presented in Figs 1-5, respectively. The recovery rate in Saudi Arabia from 1 April 2020 to 15 May 2020, the estimated PDF of the recovery rate, the estimated PDF with histogram of the recovery rate, the estimated CDF of the recovery rate and the log likelihood function of the recovery rate are presented in Figs 6-10, respectively.

Real data Application 2
A real data application is introduced to demonstrate how the proposed estimation techniques can be applied in practice. The following datasets were provided by Badar and Priest [21].
Here are the single fibers of 20 mm (dataset 1) and 10 mm (dataset 2) in gauge lengths. Dataset   The maximum likelihood estimates of the parameters for different distributions are calculated with their standard errors for the two datasets. Also, the log-likelihood function (Log L), Akaike information criteria, Bayesian information criteria and Kolmogorov-Smirnov (KS) test statistics are calculated for the two datasets. The goodness of fit for the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) is compared with the exponentiated generalized Weibull distribution (EG-W), the generalized Marshall Olkin Weibull distribution (GMO-W), the exponentiated Marshall Olkin Weibull distribution (EMO-W), the generalized Weibull distribution (G-W), the Marshall Olkin Weibull distribution (MO-W) and Weibull distribution (W). The maximum likelihood estimators (MLE) for the parameters, Log L, AIC, BIC and KS for the dataset 1 and dataset 2 are shown in Tables 8 and 9, respectively.
The best model is chosen as the one having lowest AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). The results obtained in Tables 8 and 9, indicated that the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) is a better distribution to model the datasets than the other distributions.
The maximum likelihood estimate of the reliability of the stress-strength model according to data set 1 and data set 2 is obtained as 0.458 and the 95% asymptotic confidence interval of

Conclusion
The study of the stress-strength reliability model subject to the exponentiated generalized Marshall Olkin G family of distributions is introduced. The maximum likelihood estimator, the asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are obtained. Bayesian estimators and the credible interval for the stress-strength reliability function are derived. A simulation study is introduced in which EGMO-W distribution is applied. All results obtained in the simulation study is consistent. Applications based on real data are introduced to show the results for the stress-strength model and compare the EGMO-E and EGMO-W distributions with other different distributions. These comparisons show that EGMO-E and EGMO-W distributions can be considered as better models to fit the datasets.